Chapter 12

In Problems \(7-12\), find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=x^{2} y ; x=s t, y=s-t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(z=2 e^{3 y} \cos 2 x ;(\pi / 3,0,-1)\)

Find the gradient \(\nabla f\). $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\sin \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\)

Find all first partial derivatives of each function. \(f(x, y)=\sqrt{x^{2}-y^{2}}\)

Sketch the graph of \(\bar{f}\). $$ f(x, y)=6 $$

Find the minimum distance between the origin and the plane \(x+3 y-2 z=4\)

Find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=x^{2}-y \ln x ; x=s / t, y=s^{2} t $$

Find the equation of the tangent plane to the given surface at the indicated point. \(z=x^{1 / 2}+y^{1 / 2} ;(1,4,3)\)

Find the gradient \(\nabla f\). $$ f(x, y, z)=x^{2} y+y^{2} z+z^{2} x $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{\tan \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}\)

Find all first partial derivatives of each function. \(f(u, v)=e^{u v}\)

Find a unit vector in the direction in which \(f\) increases most rapidly at \(\mathbf{p} .\) What is the rate of change in this direction? \(f(x, y)=x^{3}-y^{5} ; \mathbf{p}=(2,-1)\)

The material for the bottom of a rectangular box costs three times as much per square foot as the material for the sides and top. Find the greatest volume that such a box can have if the total amount of money available for material is \(\$ 12\) and the material for the bottom costs \(\$ 0.60\) per square foot.

Find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=e^{x^{2}+y^{2}} ; x=s \sin t, y=t \sin s $$

Use the total differential dz to approximate the change in z as \((x, y)\) moves from \(P\) to \(Q .\) Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) \(z=2 x^{2} y^{3} ; P(1,1), Q(0.99,1.02)\)

Find the gradient \(\nabla f\). $$ f(x, y, z)=x^{2} y e^{x-z} $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{x^{4}-y^{4}}\)

Find all first partial derivatives of each function. \(g(x, y)=e^{-x y}\)

Find a unit vector in the direction in which \(f\) increases most rapidly at \(\mathbf{p} .\) What is the rate of change in this direction? \(f(x, y)=e^{y} \sin x ; \mathbf{p}=(5 \pi / 6,0)\)

Find the minimum distance between the origin and the surface \(x^{2} y-z^{2}+9=0\).

Find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=\ln (x+y)-\ln (x-y) ; x=t e^{s}, y=e^{s t} $$

Use the total differential dz to approximate the change in z as \((x, y)\) moves from \(P\) to \(Q .\) Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) \(z=x^{2}-5 x y+y ; P(2,3), Q(2.03,2.98)\)

Find the gradient \(\nabla f\). $$ f(x, y, z)=x z \ln (x+y+z) $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}+y^{2}}\)

Find all first partial derivatives of each function. \(f(s, t)=\ln \left(s^{2}-t^{2}\right)\)

Find a unit vector in the direction in which \(f\) increases most rapidly at \(\mathbf{p} .\) What is the rate of change in this direction? \(f(x, y, z)=x^{2} y z ; \mathbf{p}=(1,-1,2)\)

Find the maximum volume of a closed rectangular box with faces parallel to the coordinate planes inscribed in the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 $$

Find the global maximum value and global minimum value of \(f\) on \(S\) and indicate where each occurs. \(\begin{aligned} \text f(x, y) &=3 x+4 y ; \\ S=\\{(x, y): 0&\leq x \leq 1,-1 \leq y \leq 1\\} \end{aligned}\)

Find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=\sqrt{x^{2}+y^{2}+z^{2}}, x=\cos s t, y=\sin s t, z=s^{2} t $$

find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) \((\) see Example 1). $$ f(x, y)=x^{2} y-x y^{2}, \mathbf{p}=(-2,3) $$

Use the total differential dz to approximate the change in z as \((x, y)\) moves from \(P\) to \(Q .\) Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) \(z=\ln \left(x^{2} y\right) ; P(-2,4), Q(-1.98,3.96)\)

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}}\)

Find all first partial derivatives of each function. \(f(x, y)=\tan ^{-1}(4 x-7 y)\)

Find a unit vector in the direction in which \(f\) increases most rapidly at \(\mathbf{p} .\) What is the rate of change in this direction? \(f(x, y, z)=x e^{y z} ; \mathbf{p}=(2,0,-4)\)

Find the maximum volume of the first-octant rectangular box with faces parallel to the coordinate planes, one vertex at \((0,0,0)\), and diagonally opposite vertex on the plane $$ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 $$

Find the global maximum value and global minimum value of \(f\) on \(S\) and indicate where each occurs. \(f(x, y)=x^{2}+y^{2}\); \(S=\\{(x, y):-1 \leq x \leq 3,-1 \leq y \leq 4\\}\)

Find \(\partial w / \partial t\) by using the Chain Rule. Express your final answer in terms of \(s\) and \(t\) $$ w=e^{x y+z} ; x=s+t, y=s-t, z=t^{2} $$

find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) \((\) see Example 1). $$ f(x, y)=x^{3} y+3 x y^{2}, \mathbf{p}=(2,-2) $$

Use the total differential dz to approximate the change in z as \((x, y)\) moves from \(P\) to \(Q .\) Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) \(z=\tan ^{-1} x y ; P(-2,-0.5), Q(-2.03,-0.51)\)

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\left(x^{2}+y^{2}\right)^{2}}\)

Find all first partial derivatives of each function. \(F(w, z)=w \sin ^{-1}\left(\frac{w}{z}\right)\)

Sketch the graph of \(\bar{f}\). $$ f(x, y)=\sqrt{16-4 x^{2}-y^{2}} $$

In what direction u does \(f(x, y)=1-x^{2}-y^{2}\) decrease most rapidly at \(\mathbf{p}=(-1,2) ?\)

If \(z=x^{2} y, x=2 t+s\), and \(y=1-s t^{2}\), find $$ \left.\frac{\partial z}{\partial t}\right|_{s=1, t=-2} $$

Find all points on the surface $$ z=x^{2}-2 x y-y^{2}-8 x+4 y $$ where the tangent plane is horizontal.

find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) \((\) see Example 1). $$ f(x, y)=\cos \pi x \sin \pi y+\sin 2 \pi y, \mathbf{p}=\left(-1, \frac{1}{2}\right) $$

Find the indicated limit or state that it does not exist. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{7 / 3}}{x^{2}+y^{2}}\)

Find all first partial derivatives of each function. \(f(x, y)=y \cos \left(x^{2}+y^{2}\right)\)

In what direction \(\mathbf{u}\) does \(f(x, y)=\sin (3 x-y)\) decrease most rapidly at \(\mathbf{p}=(\pi / 6, \pi / 4) ?\)